We study the behaviour of solutions with isolated singularities to weighted
$p$-Laplacian operators in the punctured unit ball centred at zero. We impose
our weight to be in the framework of regular variation. We are able to classify
the solutions to our for the entire range of $p$ by adapting the rescaling
method from Kichenassamy and V\'eronís study of isolated singularities of
$p$-harmonic functions. The adapted method relies on the construction of a
suitable "fundamental solution" depending on the range of $p$. We show that all
possible singularities at $0$ for a positive solution of our problem are either
removable (and the solution can be extended as a continuous solution in the
entire ball), weak, or the solution can be extended as a continuous function in
the entire ball. We note there exists no solutions with strong singularities to
our problem, a case which only arises when absorption terms are present.